In the books he owned a service revolver (do I understand it right that as an officer he had to purchase his own gun and that he didn't "forget" to return it to the armory when he left the army?) until it was lost in an experiment. Had he kept it and had he lived long enough, when would he have to hand it over to the police?
In addition, could Holmes or Mrs. Hudson or their milkman legally own a firearm back then, and would the cutoff date be any different for them?
The answer is, not in his lifetime, and potentially never if he moved to Northern Ireland. Also, Holmes and Mrs Hudson probably would have been allowed to own and carry firearms, though I'd be interested to know if women were restricted from getting a licence to carry a firearm in public.
Under the Gun Licences Act 1870, Watson would have required a licence to legally carry his revolver outside, but not to keep it in his house. And the Pistols Act 1903 wouldn't have affected Watson's ability to own and carry his revolver, though Holmes might have been barred as it revoked the right of "drunken or insane" people to own a gun.
The Firearms Act 1920 allowed the police to refuse a licence to anyone "unfitted to be trusted with a firearm". I think Watson would have no problems keeping his revolver under this law, which was brought in as a response to political instability in Europe, for example the Russian Revolution.
Subsequent laws - the Firearms Acts of 1937 and 1968, the Criminal Justice Act 1967 and the Firearms (Amendment) Act 1988 - didn't ban the general ownership of handguns.
It wasn't until the Firearms (Amendment) Act 1997, which in essence banned the ownership of handguns in England, Scotland and Wales, that Watson would have had to hand his service revolver in. In theory. However, there is an antique gun exemption under section 58(2) of the Firearms Act 1968, and Watson's (Second Anglo-Afghan War?) service revolver might therefore be exempted.
Also, if Watson moved to Northern Ireland (which is part of the UK) he could legally own a revolver for self defence.